Proving Quadrilaterals are Parallelograms - Sec 6.3

GOALS:To prove a quadrilateral is a parallelogram (6 ways to do so!)

Proving Quadrilaterals are Parallelograms

A quadrilateral is a parallelogram if:

1. Both pairs of opposite sides are parallel (by the definition) W X

YZ

,If WX ZY and ZW YX

then quadrilateral WXYZ is a

parallelogram

Proving Quadrilaterals are Parallelograms Theorem 6.6

A quadrilateral is a parallelogram if:

2. Both pairs of opposite sides are congruent

W X

YZ

,If WX ZY and ZW YX

then quadrilateral WXYZ is a

parallelogram

Proving Quadrilaterals are Parallelograms Theorem 6.7

A quadrilateral is a parallelogram if:

3. Both pairs of opposite angles are congruentW X

YZ

,If W Y and Z X

then quadrilateral WXYZ is a

parallelogram

Proving Quadrilaterals are Parallelograms Theorem 6.8

A quadrilateral is a parallelogram if:

4. Consecutive angles are supplementary

W X

YZ180, 180,

180, 180,

If m W X m X m Y

m Y m Z and m Z m W

then quadrilateral WXYZ is a

parallelogram

Proving Quadrilaterals are Parallelograms Theorem 6.9

A quadrilateral is a parallelogram if:

5. Diagonals bisect each other

W X

YZ,If ZM XM and WM YM

then quadrilateral WXYZ is a

parallelogram

M

Proving Quadrilaterals are Parallelograms Theorem 6.10

A quadrilateral is a parallelogram if:

6. One pair of opposite sides are congruent and parallel

W X

YZ

,If WX ZY and WX ZY

then quadrilateral WXYZ is a

parallelogram

Example

12115 65

Decide if you have enough information to determine if the following quadrilaterals are parallelograms. Explain your reasoning.

12

65

Example

W X

YZ

1

1 1

Show that W(-1, 2) X(3, 2) Y (1, -2) and Z(-3, -2) are the vertices of a parallelogram.

Example Prove Theorem 6.6

W X

YZ

:Given WX ZY and ZW YX

Prove : WXYZ is a parallelogram

Homework

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